Localized waves, which may also be referred to as non-diffractive waves, are beams and/or pulses that may be capable of resisting diffraction and/or dispersion over long distances even in guiding media. Predicted to exist in the early 1970s and obtained theoretically and experimentally as solutions to the wave equations starting in 1992, localized waves may be utilized in applications in various fields where a role is played by a wave equation, from electromagnetism extending to acoustics and optics. In electromagnetic areas, localized waves may be utilized, for instance, for secure communications, and with higher power handling capability in destruction and elimination of targets.
Localized waves include slow-decaying and low dispersing class of Maxwell's equations solutions. One such solution is often referred to as focus wave modes (FWMs). Such FWMs may be structured as three dimensional pulses that may carry energy with the speed of light in linear paths. However without an infinite energy input, finite energy solutions of a FWMs type may result in dispersion and loss of energy. To counteract such dispersion and loss of energy, a superposition of FWMs may permit finite energy solutions of a FWMs type to result in slow-decaying solutions, which may be characterized by high directivity. Such FWMs characterized by high directivity may be referred to as directed energy pulse trains (DEPTs). Another class of non-diffracting solutions to Maxwell's equations may be referred to as XWaves. Such XWaves were so named due to their shape in the plane through their axes. XWaves may travel to infinity without spreading provided that they are generated from infinite apertures. This family of Maxwell's equations solutions, including FWMs, DEPTs, and/or XWaves, thus may have an infinite total energy but finite energy density.